Summary
Let {X(t),t ≧ 0} be a stationary Gaussian process withEX(t)=0,EX 2(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t |α + o (|t|α)ast→0 for someC>0, 0<α≤2; (ii)r(t)=0(t −2γ) as t→∞ for some γ>0 and (iii) supt≥s|r(t)|<1 for eachs>0. Put ξ(t)= sup {s:0 ≦s ≦t,X(s)≧ (2logs)1/2}. The law of the iterated logarithm implies\(lim sup_{t \to \infty } (\xi (t) - t) = 0\) a.s. This paper gives the lower bound of ζ(t) and obtains an Erdős-Rèvèsz type LIL, i.e.,\(\lim \inf _{t \to \infty } (\xi (t) - t)/(t(\log t)^{{{(\alpha - 2)} \mathord{\left/ {\vphantom {{(\alpha - 2)} {(2}}} \right. \kern-\nulldelimiterspace} {(2}}\alpha )} \log \log t) = - (2 + \alpha )\sqrt \pi /(\alpha {\rm H}_\alpha (2C)^{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} )\) a.s. if 0<α<2 and\(\lim \inf _{t \to \infty } \log (\xi (t)/t)/log \log t = - 2\pi /\sqrt {2C} \). Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.
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Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati
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Shao, QM. An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes. Probab. Th. Rel. Fields 94, 119–133 (1992). https://doi.org/10.1007/BF01222513
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DOI: https://doi.org/10.1007/BF01222513